Definition:Lyapunov Function
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Definition
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Let $x_0$ be an equilibrium point of the system of differential equations $x' = \map f x$.
Then a function $V$ is a Lyapunov function of the system $x' = \map f x$ on an open set $U$ containing the equilibrium if and only if:
- $(1): \quad \map V {x_0} = 0$
- $(2): \quad \map V x > 0$ if $x \in U \setminus \set {x_0}$
- $(3): \quad \nabla V \cdot f \le 0$ for $x \in U$.
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Strict Lyapunov Function
If the inequality $(3)$ is strict except at $x_0$:
- $(3): \quad \nabla V \cdot f < 0$ for $x \in U$.
then $V$ is a strict Lyapunov function.
Also known as
The term Lyapunov function is sometimes seen spelt Liapunov function.
Also see
- Results about Lyapunov functions can be found here.
Source of Name
This entry was named for Aleksandr Mikhailovich Lyapunov.